StAIR ProjectLori Ferrington

Simplifying RadicalsAlgebra

OBJECTIVES AND STANDARDS

Simplifying Radicals• Michigan

Department of Education - High School Content Expectations

• Students will know the properties of positive and negative roots. (A1.1.2)

•  Students will know how to simplify positive and negative radicals for later use in algebraic equations. (A1.1.2)

• Students will know how to multiply radical expressions for later use in algebraic equations. (A1.1.2)

INTRODUCTIONSimplifying Radicals

• By clicking on this icon, you can access the definitions of key terms at any time during this lesson.

• This lesson is designed for Mrs. Ferrington’s Algebra class.

• You are to navigate your way through this lesson individually.

• The home page will allow you to navigate through different parts of this lesson

• If you link out to a webpage, close the window when you are done viewing the content on the webpage to return to this lesson.

• Key terms in this lesson will be shown in red.• After this lesson, you will be given an assessment to

measure your understanding.• Have Fun! :)

Click to return to the previous

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Click to return to the home

page

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WARM UPLet’s review factor trees and prime factorization.

• A prime number is only divisible by 1 and itself.

• A factor is a number that can divide another number without a remainder.

• 2 and 3 are prime factors of 6 because 23=6 and both 2 and 3 are prime.

180

10 18

3 652

32

So, we can say that the prime factorization of 180 is 22335.

WARM UPNeed extra practice with factor trees?

• Video explanation here

• Extra practice here

You think you’re ready…click the next button to continue.

WARM UP QUESTION #1Complete the factor tree below.

• Which of the following are factors of 252?

252B.) 2118

C.) 386

D.) 551

A.) 2126

WARM UP QUESTION #1Complete the factor tree below.

• Great job! 2 and 126 are factors of 252.

• Which of the following are factors of 126?

252

2 126

A.) 264

B.) 267

C.) 914

D.) 719

WARM UP QUESTION #1Complete the factor tree below.

• 2 and 126 are factors of 252.

• Fabulous! 9 and 14 are factors of 126.

• Which of the following are factors of 9?

252

2 126

9 14A.) 23

B.) 33

C.) 22

D.) 34

WARM UP QUESTION #1Complete the factor tree below.

• 2 and 126 are factors of 252.

• 9 and 14 are factors of 126.

• Excellent! 3 and 3 are factors of 9.

• Which of the following are factors of 14?

252

2 126

9 14

33A.) 23

B.) 25

C.) 37

D.) 27

WARM UP QUESTION #1Complete the factor tree below.

• 2 and 126 are factors of 252.

• 9 and 14 are factors of 126.

• 3 and 3 are factors of 9.

• Awesome! 2 and 7 are factors of 14.

252

2 126

9 14

33 72The prime factorization of 252 is 22337.Great job! Click the next arrow to try another.

WARM UP QUESTION #2Find the prime factorization of 112.

• Some common prime numbers are 2, 3, 5, 7, 11, and 13.

112A.) 2357

B.) 22227

C.) 2247

D.) 23335

WARM UP QUESTION #2Find the prime factorization of 112.

You’re right!!22227 = 112

112

2 56

8 7

42

22

Now let’s move on to some new stuff.

Great Job!

HomeSIMPLIFYING RADICALSHome

• A radical is any expression that contains a square root, cube root, etc.

• The symbol representation of a radical is √ .

• The term radical comes from the late Latin radicallis meaning “of roots” and from Latin radix meaning “root.”

We say that a radical expression is simplified, or in its simplest form, when the radicand has no square factors.

There are three parts to the following lesson that will teach you about simplifying radicals. Please follow them in order. You will be able to navigate back to review any lessons again.

I. Positive Radicals

II. Negative Radicals

III. Product of Two Radicals

Quiz

I. POSITIVE RADICALSExample #1: Simplify √12

• Check for a perfect square

• Find the prime factorization of the radicand

• Identify pairs of primes in radicand

• Simplify perfect squares

√12 = 3.4641; this is not a perfect square.

√12 = √223

= √22 √3

= √4 √3

= 2√3

12

2 6

2 3

So √12 simplifies to 2√3.

I. POSITIVE RADICALSExample #2: Simplify √162

• Be sure that any perfect squares come out in front (to the left) of the radical symbol. This prevents confusing it with numbers in the radicand.

√162 = 12.7279; this is not a perfect square.162

2 81

9 9

33 33

√162 = √23333

= √2 √33 √33

= √2 √9 √9

= 33 √2

= 9√2

I. POSITIVE RADICALSCheck for Understanding…

• Hint: If an integer is in front of the radical, do not move it.

Simplify 2√605

For extra examples go here

A.) 11√5

B.) 11√2

C.) 10√11

D.) 22√5

I. POSITIVE RADICALSCheck for Understanding…

2√605 ≠ 11√5

Don’t forget the integer in front of the radical symbol

Try Again!Oops,

I. POSITIVE RADICALSCheck for Understanding…Try Again!Sorry,2√605 ≠ 11√2

You can only square root perfect squares. Look for pairs of numbers in the radicand.

I. POSITIVE RADICALSCheck for Understanding…Try Again!Bummer,2√605 ≠ 10√11

You can only square root perfect squares. Look for pairs of numbers in the radicand.

I. POSITIVE RADICALSCheck for Understanding…Great Job!

Yes!!

Now on to negative radicals in section II …

2√605 = 2√51111

= 2 √5 √1111

= 2 √5 √121

= 211 √5

= 22√5

II. NEGATIVE RADICALSWhat do you think happens if a negative is outside of the radical?

• Hint: many of the same properties of simplifying positive radicals apply in this situation.

Let’s look at example #1: Simplify -2√72Which do you think is the correct answer?

A.) 4√6

B.) -12√2

C.) 12√3

D.) -8√3

II. NEGATIVE RADICALSWhat do you think happens if a negative is outside of the radical?

Example #1: Simplify -2√72You’re correct! The answer is B.) -12√2, but why?

What rule best fits when simplifying radicals with a negative in front of them?A.) A negative in front of the radical goes in the radicand.

B.) A negative in front of the radical stays in front.

C.) A negative in front of the radical cannot be simplified.

II. NEGATIVE RADICALSExample #2: Simplify -5√50

• Don’t forget the rule you determined in the previous slide:

A negative in front of the radical stays in front.

√50 = 7.017; this is not a perfect square.

50

2 25

5 5

-5√50 = -5√255

= -5 √2 √55

= -5 √2 √25

= -55 √2

= -25√2

II. NEGATIVE RADICALSCheck for understanding…

Simplify -2√98

For extra examples go here

A.) -14√2

B.) -7√2

C.) -2√7

D.) -4√7

II. NEGATIVE RADICALSCheck for Understanding…Try Again!Oh no,-2√98 ≠ -7√2

Don’t forget about the integer in front of the radical

II. NEGATIVE RADICALSCheck for Understanding…Try Again!Sorry,-2√98 ≠ -2√7

Look for pairs of numbers in the radicand.

II. NEGATIVE RADICALSCheck for Understanding…Try Again!Nice try,-2√98 ≠ -4√7

Look for perfect squares in the radicand.

II. NEGATIVE RADICALSCheck for Understanding…

-2√98 = -2 √277= -2 √2

√77= -2 √2

√49= -27 √2= -14√2

Fantastic!You got it!!

Let’s look at the multiplying two radicals in section III …

III. PRODUCT OF 2 RADICALS

Example #1: Simplify 2√3 √12• Check for

perfect squares.

• Multiply the numbers in front of the radical and multiply the radicands.

• Simplify the radicand by finding the factorization or by identifying perfect squares

Neither √3 nor √12 are perfect squares.

2√3 √12 = 2 √312

= 2 √36

= 26

= 12

So 2√3 √12 simplifies to 12.

III. PRODUCT OF 2 RADICALS

Example #2: Simplify 3√3 2√8

• Remember to use factor trees to help you find the prime factorization of the radicand.

Neither √3 nor √8 are perfect squares.

3√3 2√8 = 32 √38= 6 √24= 6 √2223= 6 √22 √23= 6 √4 √6= 6 2 √6= 12√6

III. PRODUCT OF 2 RADICALS

Check for understanding…

Simplify √10 √5

For extra examples go here

A.) 5√5

B.) 2√5

C.) -2√5

D.) 5√2

III. PRODUCT OF 2 RADICALS

Check for Understanding…Try Again!Oh no,√10 √5 ≠ 5√5

Look for perfect squares in the radicand.

III. PRODUCT OF 2 RADICALS

Check for Understanding…Try Again!Not that one,√10 √5 ≠ 2√5

Look for perfect squares or pairs of integers in the radicand.

III. PRODUCT OF 2 RADICALS

Check for Understanding…Try Again!Whoops,√10 √5 ≠ -2√5

Be careful of your use of negatives.

III. PRODUCT OF 2 RADICALS

Check for Understanding…

√10 √5 = √105 = √50 =

√255 = √2

√55 = √2

√25 = 5√2

Awesome!That’s correct!

Click next to take the Simplifying Radicals Quiz …

QUIZInstructions

• Don’t forget your factor trees!

• Move through this quiz by selecting the correct simplified form of the radical expression given.• You must get each problem correct

before proceeding to the next question.• Good luck!

Click here to begin the

quiz

QUIZ

QUIZ1. Simplify √200 1

A.) 20√2

B.) 10√5

C.) 10√2

D.) 2√5

QUIZ1. Simplify √200

Oops, go back and try again!

Hint: Find your prime factorization of 200, and then look for perfect squares.

QUIZ2. Simplify 10√1000 2

A.) 10√10

B.) 100√10

C.) 10√100

D.) 20√10

QUIZ2. Simplify 10√1000

Oops, go back and try again!

Hint: Don’t forget the 10 in front of the radical.

QUIZ3. Simplify -√32 3

A.) -4√8

B.) 4√4

C.) -8√2

D.) -4√2

QUIZ3. Simplify -√32

Oops, go back and try again!

Hint: Remember the rule about negatives in front of the radical symbol.

QUIZ4. Simplify -3√27 4√3 4

A.) -108

B.) -36√3

C.) -81

D.) -√93

QUIZ4. Simplify -3√27 4√3

Oops, go back and try again!

Hint: Multiply your integers in front of the radical and the multiply the radicands before you simplify.

Great!Fabulous!

You did it!Superb

Excellent

VOCABULARYTerms and definitions (in order of appearance)

• Prime number• Factor• Prime

Factorization• Radical

• Radicand• Perfect Square

A number only divisible by only 1 and itselfA number that can divide another number without a remainder.The list of all of the prime numbers whose product makes up a number.Any expression that contains a root (e.g. square root). The symbol is √The number underneath the root or radical symbol.It is the product of some integer with itself (e.g. √9 = 3).